1. The public having ignored him for many years, the writer suddenly became famous. Correct. This is an absolute phrase structure. (noun + a participle) It means that after years of being ignored, the writer unexpectedly became famous. 2. The public having ignored the writer for many years, the writer suddenly became famous. Correct. This is also an absolute phrase structure. It is more explicit because it repeats "the writer" rather than using "him." The meaning is the same as sentence (1) 3."He" in this case should not be capitalised unless it starts a new sentence. 4. Correct. It uses a subordinate clause with "though", emphasising contrast. It means that even though he was ignored for many years, he still became famous later. The meaning is similar but slightly different from sentences (1) and (2). 5.Incorrect / Awkward phrasing. Issue: "with the public having ignored him for many years" is an unnatural construction. Better version: "The writer suddenly became famous despite the public having ignored him for many years." 6. Correct. This uses a time-related structure "after" and makes the sequence of events clear. The meaning is slightly different because it focuses more on time rather than contrast.